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A new perspective on the aerodynamic performance and power1

limit of crosswind kite systems2

Mojtaba Kheiria,b,∗

, Vahid Saberi Nasrabadb, Fr´ed´eric Bourgaultb

3

aConcordia University, 1455 de Maisonneuve Blvd. West, Montr´eal, QC, H3G 1M8, Canada4

bNew Leaf Management Ltd., 500-1177 West Hastings Street, Vancouver, BC, V6E 2K3, Canada5

Abstract6

In this paper, a new perspective on the aerodynamic performance modelling of crosswind

kite power systems (CKPSs) is provided, where the eﬀects of the induction factor or ﬂow

retardation by the kite are taken into account. For simplicity, only CKPSs in straight

downwind conﬁguration are considered, where the kites sweep an area perpendicular to the

wind direction. Moreover, the in-plane or tangential induction factor is neglected. It is argued

that the concept of the swept-area-normalized power coeﬃcient, which is commonly used

for conventional wind turbines, is not practically important for CKPSs. Instead, a kite-area-

normalized power coeﬃcient concept is adopted and is shown to be a more appropriate metric

of performance of kite systems. The kite-area-normalized useful and wasted power coeﬃcients

for both a lift and a drag mode CKPS are plotted at diﬀerent values of solidity factor and

aerodynamic eﬃciency. Moreover, it is shown that the two modes of power generation, i.e. lift

and drag modes, yield the same amount of useful power when the kite system solidity factor

is essentially zero – this is in agreement with the underlying assumption in the crosswind kite

power principle. For non-zero solidity factors, thus non-zero induction factors, however, our

results suggest that the drag mode has higher power generation potential than the lift mode.

Keywords: Airborne wind energy, crosswind kite power system, induction factor,7

Betz-Joukowsky limit, actuator disc theory, crosswind principle8

∗Corresponding author. Tel.: +1 514 848 2424 ext. 4210.

Email address: mojtaba.kheiri@concordia.ca (Mojtaba Kheiri)

Preprint submitted to Journal of Wind Engineering & Industrial Aerodynamics July 22, 2019

1. Introduction9

Crosswind kite power systems (CKPSs) are a type of airborne wind energy (AWE) systems

10

which are used for harnessing high-altitude wind energy. Winds at higher altitudes are stronger

11

and steadier, thus greatly appealing for power generation. The principle of “crosswind kite

12

power” was ﬁrst introduced in a seminal paper by Loyd (1980), who showed that large

13

amounts of wind power could be harvested inexpensively by means of an aerodynamically

14

eﬃcient tethered wing (or kite) ﬂying at high speed transverse to the incoming wind direction.

15

He also proposed two diﬀerent modes for power generation, namely the lift mode (i.e. ground-

16

based generation) and the drag mode (i.e. on-board generation); see Figures 1and 2. In

17

the lift mode, power is generated by transferring mechanical power via tether tension to

18

the ground (e.g. unrolling the tether from a drum), while in the drag mode, electricity is

19

generated by on-board turbines and transmitted via conductive material in the tether to

20

the ground. Various concepts employing, e.g. ﬁxed wings, soft kites, autogyros, or airships,

21

have so far been put forward to extract high-altitude wind energy; for more details about

22

these concepts, the interested reader is referred to Fagiano and Milanese (2012); Ahrens et al.

23

(2013); Cherubini et al. (2015); Rancourt et al. (2016); Schmehl (2018).24

Nearly three decades after introduction of the crosswind kite power principle, Argatov

25

et al. (2009) proposed a “reﬁned crosswind motion law” for lift mode kites, including the

26

eﬀects of the tether drag. They also obtained a new expression for the peak value of the

27

mechanical power, taking into account the eﬀects of centrifugal, gravitational and frictional

28

forces. In a subsequent paper, Argatov et al. (2011) extended the reﬁned crosswind motion

29

law to include the eﬀects of kite’s control and gravity.30

To date, most studies on the aerodynamic performance of CKPSs have neglected the

31

eﬀects of ﬂow retardation or induction factor by a kite. The induction factor is a measure

32

of the inﬂuence of an energy harvesting device (e.g. a wind turbine) on ﬂow, and it may

33

2

be correlated to the capability of the device to harvest power from ﬂow. There have been,

34

nevertheless, a few attempts to include the eﬀects of the induction factor in the aerodynamic

35

modelling of CKPSs. For example, Zanon et al. (2014) explored ‘the impact of the airfoil-

36

airmass interaction on the extracted power’ of a single- and dual-airfoil drag mode systems.

37

They showed that, for a single large-scale kite (

∼

14MW), the extracted power is reduced by

38

27% , if in the power calculations, the airfoil-airmass interaction is also taken into account.

39

Recently, Leuthold et al. (2017) examined the relevance of axial (i.e. normal to the rotor

40

plane) and lateral (i.e. in-plane) induction factors to the modelling of lift mode multiple-kite

41

airborne wind energy systems (MAWESs). They found that the axial induction factor is

42

relevant and should be considered in the power optimization of a MAWES, while the lateral

43

induction factor may be neglected.44

Recently, Kheiri et al. (2017a,2018) developed, in a systematic manner, an extended

45

actuator disc theory that included the eﬀects of the induction factor. This theory was then

46

used to derive expressions for the potential power output from a kite in both lift and drag

47

modes. They also proposed a formula for calculating the induction factor of a CKPS. Their

48

numerical results showed that the induction factor for a crosswind kite system may, in fact,

49

be quite signiﬁcant, depending on the system parameters, in contrast to what is commonly

50

assumed. In addition, it was found that neglecting even a small induction factor may result in

51

a signiﬁcant overestimation of power output. Moreover, Kheiri et al. (2018) conducted several

52

CFD simulations to validate their theory. The CFD results showed a good agreement with

53

the theoretical results. In addition, the CFD simulations showed that a low-speed turbulent

54

wake ﬂow is formed and expands to distances several times of the gyration radius behind a

55

CKPS, quite similarly to conventional wind turbines (CWTs). Studying the wake ﬂow of a

56

single or multiple kites is essential for kite farms layout design and optimization; for some

57

details see Kheiri et al. (2017b).58

3

It is well-known that an energy extracting device, such as a wind turbine, cannot theoret-

59

ically harvest 100% of the available wind power. The amount of power than can be extracted

60

from a freestream via an energy extracting device is theoretically limited to 16/27 (or

≈

59%)

61

of the available wind power. This is commonly referred to as the Betz-Joukowsky (B-J, in

62

short) limit and may be derived from the actuator disc theory (see Okulov and van Kuik,

63

2009). Several investigators have expressed reservations about the applicability of the B-J

64

limit to crosswind kite systems; for example, refer to Loyd (1980); Archer (2013); Costello

65

et al. (2015); alternatively, the reader may refer to Kheiri et al. (2018) for a brief discussion

66

of the issue.67

Nonetheless, in a very recent paper, De Lellis et al, (2018) attempted to extend the

68

concept of the B-J limit to CKPSs. Their parallel development has resulted in the same

69

expressions as those presented in Kheiri et al. (2017a,2018) for the power output of lift

70

mode CKPSs.

1

They have argued that a horizontal-axis wind turbine (HAWT) and a drag

71

mode kite system extract power through similar means (i.e. torque in HAWT and thrust

72

of on-board turbines in drag mode CKPS). They concluded then, similarly to HAWTs, the

73

classical B-J limit should apply to a drag mode CKPS. They also showed that a lift mode

74

CKPS at best can harvest 4/27

≈

15% of the available wind power – that is 1/4 of the B-J

75

limit. In addition, they found that a lift mode CKPS at the optimal point should spend

76

exactly the same amount of power as the useful power to drive the kite.77

The present paper aims to provide a new theoretical perspective on the aerodynamic

78

modelling and power limit of CKPSs by taking into account the eﬀects of the induction factor.

79

For simplicity, only CKPSs in straight downwind conﬁguration are considered here, where the

80

kites sweep an area perpendicular to the wind direction while also reeling out (i.e. translating

81

1

This remarkable coincidence, which should perhaps not come as a surprise, raises our level of conﬁdence

in the results to be correctly representative of the physics behind the system.

4

downwind) at a constant speed. Two diﬀerent but relevant deﬁnitions of the power coeﬃcient,

82

i.e. swept- and kite-area-normalized power coeﬃcients, are provided and their applicability

83

to CKPSs will be discussed. Using the formula for obtaining the induction factor, proposed

84

by Kheiri et al. (2017a,2018), the kite-area-normalized power coeﬃcients (i.e. both useful

85

and wasted portions) are expressed as functions of independent dimensionless variables, such

86

as the solidity factor, and aerodynamic eﬃciency (refer to Section 3for deﬁnitions).87

The paper is organized as follows: ﬁrst, a few important outcomes of the extended actuator

88

disc theory are described in Section 2to serve as background reference to the rest of the paper.

89

In Section 3, expressions for the kite-area-normalized useful and wasted power coeﬃcients for

90

lift mode CKPSs are presented, and their variations as a function of the independent variables

91

(solidity factor and aerodynamic eﬃciency) are illustrated. In Section 4, the corresponding

92

power coeﬃcient expressions for drag mode CKPSs are presented and plotted. Finally, in

93

Section 5, the useful power harvesting (or power output) potential of a drag mode CKPS

94

is compared to that of a lift mode. It will be argued that a drag mode CKPS may have a

95

greater potential for useful power harvesting compared to a lift mode CKPS with the same

96

design parameters.97

2. Background98

Figure 3shows a wind turbine rotor subjected to a uniform wind ﬁeld of absolute velocity

99

v∞

and moving at constant speed

vd

in the wind direction (i.e. downwind). By deﬁning

e

as

100

the ratio of reel-out speed to freestream velocity,

vd

may be linked to

v∞

as

vd

=

ev∞

, where

101

0

≤e≤

1. Following Sørensen (2016); Kheiri et al. (2017a,2018); De Lellis et al, (2018), the

102

induction factor for a moving actuator disc or simply a rotor may be deﬁned in connection to

103

the relative incoming ﬂow velocity/wind speed, (v∞−vd), that is104

vr≡(v∞−vd)(1 −a),(1)105

5

in which vris the actual relative ﬂow velocity at the rotor, and ais the induction factor.106

It follows from the extended actuator disc theory, developed for a moving disc (refer to

107

Kheiri et al.,2018), that the absolute wake velocity is dependent on both aand e:108

vw=v∞[1 −2a(1 −e)].(2)109

Since a negative relative wake velocity, i.e.

vw−vd

, meaning a reverse wake ﬂow into the

110

moving control volume, is not sensible for an energy harvesting device, the acceptable range

111

for ais obtained by letting vw−vd≥0 which eventually leads to:112

0≤a≤1

2.(3)113

On the other hand, the axial load

2

or thrust, denoted by

T

, acting on the disc or rotor and

114

the extracted power by the rotor, P, may be written as115

T=1

2ρAsv2

∞4a(1 −a)(1 −e)2,(4)116

and117

P=T vr=1

2ρAsv3

∞4a(1 −a)2(1 −e)3,(5)118

respectively, where

As

represents the area swept by the rotor. Note that for a HAWT,

As

119

is equal to the rotor/disc area. For a CKPS, however,

As

is the annular area in the sky,

120

traversed by the kite.121

It should be noted that the total power extracted from ﬂow, that is the sum of the power

122

harvested by the rotor and that due to reeling-out, may be obtained by calculating the

123

2

This force is called axial as it is along the axis of rotation of the rotor, which is normal to the disc/rotor

plane.

6

time-rate change of the kinetic energy of ﬂow, that is124

Ptot =dK

dt =1

2˙mv2

∞−1

2˙mv2

w,(6)125

where

˙m

is the mass ﬂow rate through the actuator disc,

K

is the ﬂow kinetic energy, and

t126

is time.127

Using the expression for the wake velocity (see equation (2)), equation (6) may be

128

simpliﬁed as129

Ptot =T v∞1−a(1 −e),(7)130

in which T= ˙m(v∞−vw) has also been utilized.131

Equation (7) may be rewritten as132

Ptot =T v∞e+ (1 −a)(1 −e)=T vd+T vr,(8)133

where the ﬁrst term on the r.h.s. is the power due to reeling-out, and the second one is the

134

power harvested by the rotor (refer to equation (5)).135

One may realize that equations (4) and (5) can be obtained from the classical actuator

136

disc theory – for a stationary disc. If the expression for the thrust of a stationary disc is

137

scaled by a factor of (1

−e

)

2

and that for the power by a factor of (1

−e

)

3

, then equations

138

(4) and (5) are obtained. In other words, in the classical expressions for

T

and

P

,

v∞

should

139

simply be replaced by

v∞−vd

= (1

−e

)

v∞

, i.e. the relative velocity between the disc and

140

freestream velocity. However, one should note the diﬀerence between the deﬁnition of the

141

induction factor for a stationary disc, i.e.

vr≡v∞

(1

−a

), and that for a moving disc, given

142

in equation (1).143

7

3. Lift mode power generation144

Using the extended actuator disc and the blade element momentum theories, and neglecting

145

some high-order terms with negligible contributions, the induction factor,

a

, for a lift mode

146

CKPS in the straight downwind conﬁguration may be written as (refer to Kheiri et al.,2018)

147

a

1−a=1

4(Ak

As

)CL(CL

CD

)2,(9)148

where

Ak

and

As

are, respectively, the planform area of the kite and the area (e.g. annulus)

149

swept by the kite in the sky;

CL

and

CD

are the kite aerodynamic lift and drag coeﬃcients,

150

respectively. In general,

CD

also includes the normalized or equivalent drag coeﬃcient of the

151

tether, CDt, in addition to the drag coeﬃcient of the kite, CDk, i.e. CD=CDk+CDt.152

In equation (9), the area ratio

σ

=

Ak/As

may, in fact, be called the solidity factor of

153

the kite system, in accordance with the terminology used for wind turbines – total blade

154

area divided by the rotor disc area (see Burton et al.,2011). It can easily be concluded from

155

the equation that, increasing either the solidity factor,

CL

and/or (

CL/CD

) will increase the

156

induction factor. It should be noted that this equation is valid only for small angles of attack,

157

very large values of the lift-to-drag ratio, i.e. (

CL/CD

)

>>

1, and a straight downwind

158

conﬁguration. It is recalled that the straight downwind conﬁguration corresponds to a system

159

with the tether aligned with the wind and the kite sweeping an annulus perpendicular to the

160

wind ﬂow direction.161

In addition,

χ

=

CL

(

CL/CD

)

2

may be called the aerodynamic eﬃciency. As discussed

162

at the end of this section, a higher aerodynamic eﬃciency enhances the power generation

163

eﬃciency. It is also interesting to note that for an aircraft in the steady level ﬂight, minimum

164

power required occurs when χ1/2= (C3/2

L/CD) is maximum (refer to Anderson,1999).165

On the other hand, it can be shown that there is a correlation between the lift-to-drag

166

8

ratio, (

CL/CD

), or alternatively, the aerodynamic eﬃciency, and the tip-speed-ratio of the kite

167

system,

λ

, which may be deﬁned as the ratio of the crosswind speed

vc

to the undisturbed

168

relative wind velocity, v∞(1 −e), that is169

λ=vc

v∞(1 −e)=4(CL/CD)

4 + σCL(CL/CD)2=χ1/2

C1/2

L(1 + 1

4σχ),(10)170

where it is recalled that

e

is the ratio of reel-out speed,

vd

, to freestream velocity,

v∞

, i.e.

171

vd

=

ev∞

; also,

vc

= (

CL/CD

)(1

−e

)(1

−a

)

v∞

; refer to Kheiri et al. (2018) for details; also,

172

see Figure 4.3

173

Taking also into account the induction factor, the expression for the useful/net power

174

harvested via a kite ﬂying steadily in lift mode during the power generation phase of a

175

pumping cycle (i.e. the cycle of reel-out or power generation phase and reel-in or power

176

consumption phase; see Figure 1) may be obtained by multiplying the thrust acting on the

177

rotor area (refer to equation (4)) by the reel-out speed, as178

PL=T vd=1

2ρAsv3

∞4a(1 −a)(1 −e)2e, (11)179

We may deﬁne a power coeﬃcient by normalizing the useful harvested power (or power

180

output) by the wind power available to an area equal to As, as follows181

C(s)

pL=PL

1

2ρAsv3

∞

= 4a(1 −a)(1 −e)2e, (12)182

where the superscript (

s

) indicates that the power has been normalized with respect to the

183

wind power available to the swept area,

As

. This is a standard deﬁnition for the power

184

coeﬃcient for conventional wind turbines (CWTs).185

3It can easily be concluded that for σCL(CL/CD)2=σχ << 1 or a << 1, λ'(CL/CD).

9

From equation (12) and assuming that

a

and

e

are independent variables,

4

one may easily

186

ﬁnd that the maximum value of C(s)

pLoccurs when e= 1/3 and a= 1/2:187

C(s)

pL,max =4

27,(13)188

which is consistent with the results obtained by De Lellis et al, (2018).189

It is well-known that for a CWT, the maximum power coeﬃcient, according to the

190

Betz-Joukowsky limit is 16/27, which is four-times of the

C(s)

pL,max

. The power coeﬃcient is,

191

however, only one of the performance factors. It is reminded that the premise of a CKPS is

192

to access stronger high-altitude winds when also sweeping a much larger area in the sky with

193

less material, compared to a CWT. In fact, the cumulative impact of the latter factors is

194

believed to outweigh the impact of the power coeﬃcient, making CKPSs very appealing in

195

terms of the Levelized Cost of Energy (LCOE).196

From equation (9) and using deﬁnitions of

σ

and

χ

, in order to achieve

a

= 1

/

2 with a

197

lift mode kite, we should have198

1

4σχ = 1.(14)199

Then assuming a practical value for

χ

, say 100 (

CL

= 1

, CD

= 0

.

1), it follows that the

200

solidity factor should be 0.04 which is quite large for a CKPS, and it is, in fact, comparable

201

to the solidity factor of a typical, modern three-blade CWT (refer to Burton et al.,2011).

202

For a single kite system, this large solidity factor corresponds to a circular ﬂight trajectory of

203

a quite small gyration radius. Flying in tight loops creates large centrifugal forces, which

204

4

In principle, the question is whether wind turbine’s power extraction capability depends on the magnitude

of the incoming wind. In reality, the magnitude of the ﬂow velocity controls the Reynolds number which

aﬀects aerodynamic lift and drag coeﬃcients and in turn the power extraction capability. However, depending

on how much the incoming ﬂow velocity is varied, the eﬀects on the aerodynamic eﬃciency and thus the

power extraction capability may be negligible. For example, Laursen et al. (2007)’s CFD simulation results

for a multi-Megawatt CWT conﬁrm that the axial induction factor changes very little when the freestream

velocity changes from 6 m/s to 10 m/s.

10

have to be sustained by the kite structure, and is also challenging from the ﬂight control

205

point of view.206

One possible way to achieve large solidity factors while not creating large centrifugal forces

207

or hindering maneuverability and aerodynamic eﬃciency may be to adopt a multiple-kite

208

system concept. One such concept is what is commonly referred to as “dancing kites” or

209

“dual-airfoil”; see, for example, Payne (1976); Houska and Diehl (2007); Zanon et al. (2013);

210

Cherubini (2017). The dancing kites or the dual-airfoil concept was originally put forward to

211

reduce the losses due to the tether drag. In this concept, the individual kites are connected

212

via shorter secondary tethers ﬂying at high speed to the longer main tether which remains

213

nearly stationary. Nevertheless, the need for a carefully designed ﬂight control system and

214

trajectory planning make this concept challenging.215

Higher values of

χ

would obviously result into lower required values of

σ

– larger

As

for

216

the same

Ak

. An eﬃcient wing with low drag and high lift is thus desirable, but note that

217

CDalso includes the equivalent tether drag coeﬃcient which may not be easily reduced.218

An alternative, and also practically more satisfactory, deﬁnition of power coeﬃcient for

219

CKPSs is proposed, by normalizing the power output by the wind power available to an

220

airborne area equal to the kite planform area,

Ak

. This deﬁnition is similar to the “Power

221

Harvesting Factor

ζ

” put forward by Diehl (2013). It is argued that based on the crosswind

222

kite power principle (refer to Loyd,1980), power is harvested by heavily loading the kite

223

aerodynamically, where the kite planform area plays an important role. In contrast, for

224

CWTs, the loading and power generated correlate to the rotor diameter and thus to the swept

225

area. This may be clariﬁed further by comparing the axial load/thrust per unit lifting surface

226

(kite or blade) area for a typical lift mode CKPS with that for a typical CWT. Alternatively,

227

one may compare the thrust coeﬃcients obtained by normalizing the actual axial load by the

228

11

force due to freestream dynamic pressure, as (refer to equation (4))229

C(k)

T=T

1

2ρAkv2

∞

=4

σa(1 −a)(1 −e)2,(15)230

in which the superscript (

k

) indicates that the axial load/thrust has been normalized with

231

respect to the lifting surface area –

Ak

is used here to refer to both the kite and blade

232

planform area.233

For a typical CWT with

e

= 0,

σ

= 0

.

04 and

a

= 1

/

3 (to get the maximum power),

234

equation (15) yields C(k)

TCW T ≈22.2.235

In order to obtain the thrust coeﬃcient value for a typical lift mode CKPS, equation (15)

236

should be simpliﬁed further via equation (9), as237

C(k)

T(σ, χ, e) = χ

(1 + 1

4χσ)2(1 −e)2.(16)238

As it will be discussed in the following paragraphs, the peak value of the kite-area-normalized

239

power coeﬃcient for a CKPS is achieved when the solidity factor is very small, or essentially

240

zero. Thus, for a typical lift mode CKPS with

e

= 1

/

3,

σ

= 0 and

χ

= 100 (where the

241

power output will be the same as that from the CWT considered above), equation (16) yields

242

C(k)

TCK P S ≈

44

.

4, which is exactly 2 times

C(k)

TCW T

. In addition, if the CWT has three blades

243

with the total planform area equal to that of the kite (in order the CWT and the kite to

244

have the same airborne area), then the axial load acting on the kite will be 3

×

2 = 6 times

245

the load on each of the blades of the CWT.246

Therefore, it appears reasonable to deﬁne the useful power coeﬃcient for a lift mode

247

CKPS as follows (refer to equation (11)),248

C(k)

pL=PL

1

2ρAkv3

∞

=4

σa(1 −a)(1 −e)2e. (17)249

12

In other words,

C(k)

pL

determines how much net power can be harvested in lift mode per unit

250

of planform area of the airborne kite.251

It is instructive to obtain C(k)

pLas a function of independent variables σ,χ, and e. Using252

equation (9),

a

can be written in terms of

σ

and

χ

; thus, equation (17) may be re-written as

253

C(k)

pL=C(k)

pL(σ, χ, e) = χ

(1 + 1

4χσ)2e(1 −e)2,(18)254

which conﬁrms that the maximum value of

C(k)

pL

, i.e.

C(k)

pL,max

= (4

/

27)

χ

, occurs when

e

= 1

/

3

255

and

σ

= 0, meaning that to generate the maximum power, a lift mode kite with a ﬁnite

256

planform area should spin in an inﬁnitely large area in the sky while it is reeled-out at 1

/

3

257

of the undisturbed wind speed. This conclusion is consistent with Loyd (1980)’s and Diehl

258

(2013)’s observations.259

Figure 5shows the variation of

C(k)

pL

as a function of

σ

and

χ

for

e

= 1

/

3. As seen,

C(k)

pL

260

increases as

χ

is increased and

σ

is decreased – the optimum point lies on the

σ

= 0 line. In

261

this ﬁgure and all subsequent ﬁgures for the lift mode system, the black polyline separates

262

acceptable solution points (i.e. inner region) from unacceptable solution points (i.e. outer

263

region). The acceptable region was obtained by considering

aL≤

1

/

2, where the subscript

264

L

is used to indicate the induction factor for a lift mode CKPS. As previously explained in

265

Section 2, this range is found by considering the fact that the relative velocity of the wake

266

ﬂow should not be negative; please refer to equation (3).267

Figure 6shows the variation of the induction factor as a function of

σ

and

χ

for

e

= 1

/

3.

268

As expected, by increasing the solidity factor and the aerodynamic eﬃciency (which may be

269

linked to the tip-speed-ratio using equation (10)), the induction factor increases.270

On the other hand, it is observed that some power is spent to drag the kite at high-speed

271

in the crosswind direction. This is, in fact, the wasted portion of the total power extracted

272

from wind ﬂow. The expression for the power loss is obtained by multiplying the component

273

13

of the drag force in the crosswind direction by the crosswind speed (see Figure 4), as follows

274

PL,loss =1

2ρAkV2

relCDcos αvc,(19)275

where Vrel is the relative ﬂuid-kite velocity, and αis the angle of attack.276

Assuming

Vrel 'vc

, and

cos α'

1 and after some mathematical operation, the coeﬃcient

277

of the power loss may be obtained as278

C(k)

pL,loss =C(k)

pL,loss (σ, χ, e) = χ

(1 + 1

4χσ)3(1 −e)3.(20)279

It is evident from equation (20) that

C(k)

pL,loss

(

σ

= 0

, χ, e

= 1

/

3) = (8

/

27)

χ

= 2

C(k)

pL,max

, meaning

280

that two times of the useful power will be wasted (in form of drag losses) at the optimal

281

point of the lift mode power generation. This is a surprising result, but as also mentioned by

282

Diehl (2013), it may be explained by considering the fact that the optimal point for useful

283

power does not necessarily coincide with that for eﬃciency – if we deﬁne the eﬃciency as the

284

ratio of the useful power to the useful+wasted power, i.e.

η

=

e/e

+ (1

−e

)(1 + 1

/

4

χσ

)

−1

;

285

refer to equations (18) and (20). As seen from the equation, by increasing χ,ηis increased,286

and it follows that a 100% eﬃciency is achieved in the limit of

χ→ ∞

, meaning that the

287

kite should spin at inﬁnitely high speed. However, spinning at such speeds yields zero useful

288

power (refer to equation (18)) – the rotor functions like a perfect blockage. Figure 7shows

289

variation of

C(k)

pL,loss

as a function of the solidity factor

σ

and the aerodynamic eﬃciency

χ

.

290

It can easily be veriﬁed that for non-zero

σ

and

χ

, and for

e

= 1

/

3, the ratio between the

291

power loss and the useful power coeﬃcients would be less than 2.292

One interesting observation is that the amount of wasted power in lift mode is exactly the

293

same as the power harvested by the rotor (see equation (5)) – the proof is left to the reader.

294

In other words, from the total power harvested by the kite system in lift mode, the power

295

14

due to reeling-out is the useful part and that extracted by the rotor is the wasted part.296

4. Drag mode power generation297

The induction factor for a drag mode CKPS, assuming that the drag due to on-board

298

turbines acts in the same direction as the kite drag, may be written as (refer to Kheiri et al.,

299

2018)300

a

1−a=1

4(Ak

As

)CL(CL

CD

)2(1

1 + κ)2,(21)301

where

κ

is the ratio of the drag/thrust of turbines to the kite drag, i.e.

κ

= (

Dp/D

) =

302

(

CDp/CD

),

Dp

and

CDp

being, respectively, the drag due to the turbines and the corresponding

303

coeﬃcient.304

It can be shown that the same expression as in equation (21) is obtained if the drag due

305

to the turbines acts in the crosswind direction, subject to the key assumptions made in the

306

present analysis. That conﬁguration, in fact, mimics the mechanism by which a conventional

307

wind turbine produces power. It is evident from equation (21) that the induction factor of a

308

drag mode kite, in addition to the solidity factor,

σ

= (

Ak/As

), and the kite aerodynamic

309

eﬃciency,

χ

=

CL

(

CL/CD

)

2

, is also dependent on the factor

κ

. As seen, by increasing

κ

(or

310

the thrust of turbines), the induction factor

a

will decrease. Increasing

κ

will reduce the

311

‘eﬀective’ (i.e. kite-turbine system) aerodynamic eﬃciency

χ∗

=

χ/

(1 +

κ

)

2

which in turn

312

correlates with a higher angle of attack and a lower rotational speed (or crosswind speed). A

313

lower rotational speed means a lower tip-speed-ratio and thus a smaller induction factor.314

The expression for the useful harvested power (or power output) via a kite in drag mode

315

in its general form may be obtained by multiplying the thrust of turbines by the crosswind

316

speed; the ﬁnal expression after several mathematical operation may be written as317

PD=1

2ρAkv3

∞CL(CL

CD

)2(1 −a)3κ

(1 + κ)3.(22)318

15

Equation (21) is used to write

a

as a function of

σ

,

χ

, and

κ

; thus, the kite-area-normalized

319

power coeﬃcient for a drag mode CKPS may be written as320

C(k)

pD(σ, χ, κ) = PD

1

2ρAkv3

∞

=χ

(1 + κ)2+1

4χσ3κ(1 + κ)3.(23)321

It can be easily shown from equation (23) that the maximum value of

C(k)

pD

, i.e.

C(k)

pD,max

=

322

(4

/

27)

χ

, occurs when

κ

= 1

/

2 and

σ

= 0, meaning that, to get the maximum useful power in

323

drag mode, a kite with a ﬁnite planform area should spin in an inﬁnitely large area in the

324

sky while it is equipped with a number of turbines, whose total thrust is 50% of the kite

325

system drag. As discussed in Section 3, the maximum kite-area-normalized power coeﬃcient

326

for a lift mode CKPS was

CpL,max

= (4

/

27)

χ

, then conﬁrming that a CKPS with a negligible

327

solidity factor would generate the same maximum power in lift and drag modes; a similar

328

conclusion is inferred from Loyd (1980).329

To obtain the maximum value of

C(k)

pD

for non-zero

σ

and

χ

, we need to ﬁnd the optimal

330

κfrom ∂C (k)

pD/∂κ = 0 which yields the following cubic polynomial331

−2κ3−3κ2+ 4Cκ +C+ 1 = 0,(24)332

where C=1

4σχ.333

The cubic equation (24) can have up to three real roots. However, it can be shown

334

mathematically that only one of the roots is positive; see Appendix A for more details.335

Figure 8shows the variation of the optimal

κ

, denoted by

κ∗

, as a function of

σ

and

χ

.

336

As seen from the ﬁgure, for non-zero σand χ, by increasing either σor χ,κ∗is increased.337

Figure 9shows the variation of

C(k)

pD

as a function of

σ

and

χ

for

κ

=

κ∗

. Similarly to the

338

lift mode (refer to Figure 5),

C(k)

pD

increases as the solidity factor,

σ

, is decreased and the

339

aerodynamic eﬃciency, χ, is increased.340

16

Figure 10 shows the variation of the induction factor,

aD

, as a function of the solidity

341

factor,

σ

, and the aerodynamic eﬃciency,

χ

. The subscript

D

indicates the induction factor for

342

a drag mode CKPS. It is reminded that the local optimal

κ

is always used in the calculations.

343

As expected, by increasing either

σ

or

χ

, the induction factor is increased. It is found from

344

Figures 6and 10 that,

aD

is several times smaller than

aL

for the same

σ

and

χ

. However, a

345

direct comparison between numerical values of

aD

and

aL

may not be very sensible as

aL

is

346

deﬁned in connection to the relative wind speed,

v∞

(1

−e

), at the rotor, while

aD

is deﬁned

347

in connection to the absolute wind speed, v∞.348

For drag mode CKPS, similarly to the lift mode system, some power is spent/wasted to349

drag the kite and the on-board turbines at high-speed in the crosswind direction. Recalling

350

that the kite drag is 1

/κ

times the thrust of turbines, the kite-area-normalized wasted power

351

in drag mode can easily be found by dividing C(k)

pD(refer to equation (23)) by κ:352

C(k)

pD,loss =C(k)

pD,loss (σ, χ, κ) = χ

(1 + κ)2+1

4χσ3(1 + κ)3.(25)353

It is found from equation (25) that

C(k)

pD,loss

(

σ

= 0

, χ, κ

= 1

/

2) = (8

/

27)

χ

= 2

C(k)

pD,max

. In other

354

words, twice the amount of the useful power has to be spent in order to spin the kite within

355

the annulus; the same conclusion was made for the lift mode system. Figure 11 shows the

356

variation of C(k)

pD,loss as a function of the solidity factor, σ, and the aerodynamic eﬃciency, χ.357

It can easily be veriﬁed that the ratio of the wasted power to the useful power for non-zero

358

solidity factors is always less than 2.359

The useful/net power generated in the drag mode can also be described in terms of the

360

swept area, As. Using equations (21) and (22) yields361

PD=1

2ρAsv3

∞4a(1 −a)2κ

1 + κ.(26)362

17

Similarly to the lift mode, for a drag mode system, a power coeﬃcient may be obtained by

363

normalizing PDby the wind power available to the swept area, as follows364

C(s)

PD=PD

1

2ρAsv3

∞

= 4a(1 −a)2κ

1 + κ,(27)365

in which it is reminded that the superscript (s) denotes normalization with respect to As.366

As seen from equation (27), to maximize

C(s)

pD

,

κ

should be inﬁnitely large, i.e.

κ→ ∞

,

367

and

a

= 1

/

3. Theoretically, a very large

κ

may be achieved via an aerodynamically ideal

368

wing with essentially zero drag. The maximum

C(s)

pD

then becomes

C(s)

pD,max

= 16

/

27, which is

369

equal to the B-J limit and four times the

C(s)

pL,max

. Nevertheless, as previously suggested, the

370

swept-area-normalized power coeﬃcient, such as

C(s)

pD

may not be practically signiﬁcant for

371

CKPS although it is a key performance factor for CWTs.372

5. Discussion373

Figure 12 gives an interesting plot showing the ratio of the kite-area-normalized useful

374

power coeﬃcient for the drag mode to that for the lift mode as a function of the solidity factor,

375

σ

, and the aerodynamic eﬃciency,

χ

. For all the points in the plot, it has been assumed

376

that

e

= 1

/

3 for the lift mode and

κ

=

κ∗

for the drag mode CKPS. As seen from the ﬁgure,

377

the drag mode generally outperforms the lift mode in terms of power generation (they are

378

equivalent only on the

σ

= 0 line), and this superiority becomes more evident as

σ

and

χ

are

379

increased. This is an important fundamental ﬁnding as, until now, the general perception

380

has been that the two modes are equivalent in terms of power generation potential, most

381

likely because the eﬀects of induction factor has always been neglected – only

σ

= 0 line was

382

discernible.383

However, for practical reasons, spinning in an inﬁnitely large area is not viable, and as a

384

result the solidity factor and the induction factor of a CKPS are expected to be perceptible.

385

18

The reason why the out-performance of drag mode over lift mode has been left seemingly

386

unnoticed in practice, at least so far, may be explained in part by the fact that most

387

experimental prototypes are small-scale kites ﬂying over relatively large annuli/capture areas

388

in the sky (i.e. small solidity factors), thus leading to very small induction factors. For

389

example, using the values given in (Vander Lind,2013, Fig. 28.7) for the Wing 7 prototype

390

of Makani Power (rated average power: 20 kW, rated wind speed: 10 m/s) and assuming the

391

simpliﬁed straight downwind conﬁguration, the solidity factor and the aerodynamic eﬃciency

392

of the kite are found as

σ≈

0

.

0016 and

χ

= 128, respectively. If it is further assumed that the

393

thrust/drag of on-board turbines is half of the kite drag (i.e.

κ

= 1

/

2), then using equation

394

(21), the induction factor for the Wing 7 prototype becomes

a≈

0

.

02; also,

C(k)

pD/C(k)

pL≈

1

.

03.

395

In contrast, as also discussed in Kheiri et al. (2018), for large-scale systems the induction

396

factor may be appreciable and its eﬀects, if neglected, will result into signiﬁcant overestimation

397

of the power output. This can in turn negatively aﬀect cost estimation and examination of

398

economical prospects of those systems. For example, for a 5.5 MW kite system with the

399

solidity factor

σ

= 0

.

0048, it was shown analytically as well as computationally that the

400

induction factor was of the order of 10% which if neglected, it would result into over 20%

401

power output overestimation, which is quite signiﬁcant.402

After all, Figure 12 does not demonstrate the total superiority of the drag mode over the

403

lift mode power generation. In order to draw such conclusions, it is essential to include the

404

estimation of costs incurred by the two modes of power generation, as well as to use a more

405

reﬁned theory which takes into account, e.g., the tether drag, the wind speed variation versus

406

altitude, non-ideal inclined conﬁgurations of CKPSs, and the pumping cycle of the lift mode

407

power generation. As an example, tethers for on-board or drag mode generation concepts,

408

compared to those used for ground-based or lift mode generation, are generally thicker –

409

which means higher drag force – and heavier as they contain conductive wires to transmit

410

19

electricity to the ground station (Dunker,2018). Thicker and heavier tethers reduce power

411

output and increase the cost. In addition to heavier tethers, on-board generation systems

412

carry extra mass due to turbines and voltage converters. In fact, what rules in the end is

413

the levelized cost of energy and not the power generation potential. Obtaining the levelized

414

cost of energy for the two modes is, however, beyond the scope of the present paper and is

415

deferred to a future publication.416

We close this section by highlighting the diﬀerence between two design optimization

417

philosophies that may be adopted for CKPSs and more generally for AWE systems. See

418

Figure 13 which shows the variation of the kite-area-normalized useful power coeﬃcient

419

divided by the aerodynamic eﬃciency, (

C(k)

pL/χ

), and the swept-area-normalized useful power

420

coeﬃcient,

C(s)

pL

, for a lift mode CKPS as a function of the induction factor,

aL

. It is assumed

421

that the reel-out speed ratio is 1/3. It is further assumed that the system initially has the

422

solidity factor

σ0

, the aerodynamic eﬃciency

χ0

and the corresponding induction factor

aL0

.

423

This design point is marked on the kite- and swept-area-normalized power coeﬃcient curves

424

by points (1) and (2), respectively.425

As seen from the ﬁgure, it is possible to reach the local maximum power coeﬃcient, by

426

reducing the induction factor to essentially zero on the

aL−

(

C(k)

pL/χ

) curve, or by increasing

427

the induction factor to 1/2 on the

aL−C(s)

pL

curve. Assuming that the aerodynamic eﬃciency

428

remains constant, the former may be achieved by decreasing the solidity factor via expanding

429

the swept area, while the latter may be reached by increasing the solidity factor via enlarging

430

the kite planform area. In other words, in the ﬁrst scenario, the optimization philosophy

431

is to keep the kite area constant and to increase the swept area, while that in the second

432

situation is to keep the swept area constant and to enlarge the kite area. From arguments

433

in Section 3, and also recalling that the premise of the AWE is to harvest high-altitude

434

winds, it appears more reasonable that the ﬁrst philosophy be adopted by AWE systems

435

20

designers, while the second one by the CWT design community. It should be noted that

436

in both scenarios, a stronger tether is required to sustain higher tensions produced at the

437

local maxima. In addition, the ﬁrst scenario incurs more lengths of tether, given that the

438

tether elevation angle remains constant, to allow for a larger swept area. This would in turn

439

add more mass to the system and would increase the total drag. On the other hand, in the

440

second scenario a larger kite is required, meaning larger airborne mass.441

Similarly, Figure 14 shows the curves for useful power coeﬃcient (both the kite-area-

442

normalized divided by the aerodynamic eﬃciency and the swept-area-normalized) versus

443

induction factor for a drag mode CKPS. For this mode also, as seen from the ﬁgure, the

444

optimal points for the two curves do not coincide. They occur at

aD

= 0 and

aD

= 1

/

3

445

with the power coeﬃcient values of 4

/

27 and 16

/

27, respectively (cf. the optimal points

446

for lift mode CKPS). It is also interesting to see that both

C(k)

pD/χ

and

C(s)

pD

vary almost

447

linearly with

aD

(cf. Figure 13). As seen from the ﬁgure, to maximize the power output

448

coeﬃcient for a drag mode kite with the initial induction factor

aD0

, one may adopt one of

449

the following two approaches: (1) decreasing

aD

essentially to zero by expanding the swept

450

area, (2) increasing

aD

to 1

/

3 by increasing the kite planform area. As discussed above, the

451

ﬁrst approach seems to be favoured by the AWE community, while the second approach

452

sounds to be aligned with the design philosophy of CWTs. Finally, it should be emphasized

453

that comparing directly the optimal value of the kite-area-normalized power coeﬃcient with

454

that of the swept-area-normalized power coeﬃcient is meaningless (as they are two diﬀerent

455

normalized forms of power output), and thus one being higher than the other does not mean

456

the corresponding design optimization philosophy has to be favoured.457

21

6. Concluding Remarks458

In this paper, expressions for useful (or usable) and wasted portions of power extracted

459

by lift and drag mode crosswind kite power systems, taking into account the induction

460

factor, were presented. These expressions had been obtained using the extended actuator disc

461

and blade element momentum theories. Two possible descriptions of the power coeﬃcient,

462

namely the kite-area-normalized and swept-area-normalized, were given. It was discussed

463

that, in contrast to conventional wind turbines, for crosswind kite power systems, the swept-

464

area-normalized power coeﬃcient does not properly quantiﬁes the aerodynamic performance.

465

Alternatively, the kite-area-normalized power coeﬃcient was proposed to be used for CKPSs.

466

The power coeﬃcient for a lift mode CKPS was written as a function of the solidity factor,

467

aerodynamic eﬃciency, and the reel-out ratio, while that for a drag mode system was given as

468

a function of the ﬁrst two variables plus the ratio of the on-board turbines thrust to kite drag.

469

It was shown that, for both modes and for a given value of the aerodynamic eﬃciency, the

470

maximum useful power is achieved when the solidity factor or essentially the induction factor

471

is zero – this is equivalent to a kite of a ﬁnite planform area spinning in an inﬁnitely large

472

area. In other words, the kite-area-normalized power coeﬃcient is increased by increasing

473

the aerodynamic eﬃciency and by decreasing the solidity factor. Moreover, it was conﬁrmed

474

that the lift and drag mode CKPSs can harvest the same amount of useful power when the475

solidity factor (or the induction factor) is negligibly small, where also two times of the useful

476

power is spent (or wasted) to spin the kite. For non-zero induction factors, however, it was

477

found that the drag mode outperforms the lift mode in terms of potential power output.

478

Nevertheless, what matters in the end is the levelized cost of energy which can be calculated

479

more accurately using the expressions presented in this paper.480

22

Acknowledgments481

The ﬁnancial support from New Leaf in the course of this research is sincerely appreciated.

482

The ﬁrst author is also grateful to the Gina Cody School of Engineering and Computer Science

483

of Concordia University for a start-up research grant and to the Oﬃce of Vice-President,

484

Research and Graduate Studies for an Individual Seed Program funding.485

23

Appendix A: Roots of the cubic equation used for obtaining the optimal value486

of the ratio of turbines thrust to kite drag – equation (24)487

The cubic equation for ﬁnding the optimal value of

κ

has only one positive real root. This

488

can be shown mathematically, at least, for cases where C << 1.489

We can re-write equation (24) as follows:490

(κ−r)(−α1κ2−α2κ−α3) = 0,(A.1)491

where α1= 2, α2= 3 + 2r,α3=r(3 + 2r)−4C, and κ1=r > 0 is the only positive root.492

The roots of the quadratic term in equation (A.1) are obtained as493

κ2,3=α2±√∆

−2α1

,(A.2)494

in which ∆ = α2

2−8rα2+ 32C.495

Since

r >

0 and

C <<

1, then ∆

< α2

2

and in turn

√∆< α2

. This will lead to

496

α2−√∆>

0 and also

α2

+

√∆>

0. As a result, from equation (A.2) and since

α1>

0 both

497

κ2

and

κ3

will be negative. Figure 15 shows the variation of the three roots of the cubic

498

equation, obtained numerically, for a wide range of values of C.499

24

References500

Ahrens, U., Diehl, M., Schmehl, R. (Eds.), 2013. Airborne Wind Energy. Berlin: Springer.501

Anderson, J. D., 1999. Aircraft Performance and Design. WCB McGraw-Hill.502

Archer, C., 2013. An introduction to meteorology for airborne wind energy. In: Ahrens U.,

503

Diehl M., Schmehl R., editors. Airborne Wind Energy. Berlin Heidelberg: Springer, 81–94.

504

Argatov, I., Rautakorpi, P., Silvennoinen, R., 2009. Estimation of the mechanical energy

505

output of the kite wind generator. Renewable Energy 34, 1525–1532.506

Argatov, I., Rautakorpi, P., Silvennoinen, R., 2011. Apparent wind load eﬀects on the tether

507

of a kite wind generator. Journal of Wind Engineering and Industrial Aerodynamics 99,

508

1079–1088.509

Burton, T., Jenkins, N., Sharpe, D., Bossanyi, E., 2011. Wind Energy Handbook. John

510

Wiley & Sons.511

Cherubini, A., Papini, A., Vertechy, R., Fontana, M., 2015. Airborne wind energy systems:

512

A review of the technologies. Renewable and Sustainable Energy Reviews 51, 1461–1476.513

Cherubini, A., 2017. Advances in Airborne Wind Energy and Wind Drones. PhD Thesis,

514

Scuola Superiore Sant’Anna, Pisa, Italy.515

Costello, S., Costello, C., Fran¸cois, G., Bonvin, D., 2015. Analysis of the maximum eﬃciency

516

of kite-power systems. Journal of Renewable and Sustainable Energy 7, 053108.517

Daidalos Capital GmbH, 2016. Wind Drone Technology.

https://daidalos-capital.com

,

518

[Online; accessed 12-April-2019].519

De Lellis, M., Reginatto, R., Saraiva, R., Troﬁno, A., 2018. The Betz limit applied to

520

Airborne Wind Energy. Renewable Energy 127, 32–40.521

25

Diehl, Moritz, 2013. Airborne wind energy: Basic concepts and physical foundations. In:

522

Ahrens U., Diehl M., Schmehl R., editors. Airborne Wind Energy. Berlin Heidelberg:

523

Springer, 3–22.524

Dunker, Storm, 2018. Tether and bridle line drag in airborne wind energy applications. In:

525

Schmehl R., editor. Airborne Wind Energy. Singapore: Springer, 29–56.526

Fagiano, L., Milanese, M., 2012. Airborne wind energy: An overview. American Control

527

Conference (ACC) 3132–3143, IEEE.528

Houska, B., Diehl, M., 2007. Optimal control for power generating kites. In: 2007 European

529

Control Conference (ECC), IEEE, 3560–3567.530

Kheiri, M., Bourgault, F., Saberi Nasrabad, V., 2017a. Power limit for crosswind kite systems.

531

In: Book of Abstracts of International Symposium on Wind and Tidal Power (ISWTP)

532

2017, Montr´eal, Canada, May 28-30.533

Kheiri, M., Saberi Nasrabad, V., Victor, S., Bourgault, F., 2017b. A wake model for crosswind

534

kite systems. In: Book of Abstracts of Airborne Wind Energy Conference (AWEC) 2017,535

Germany, October 5-6.536

Kheiri, M., Bourgault, F., Saberi Nasrabad, V., Victor, S., 2018. On the aerodynamic

537

performance of crosswind kite power systems. Journal of Wind Engineering and Industrial

538

Aerodynamics 181, 1-13.539

Laursen, J., Enevoldsen, P., Hjort, S., 2007. 3D CFD quantiﬁcation of the performance of a

540

multi-megawatt wind turbine. Journal of Physics: Conference Series 75(1), 012007.541

Leuthold, R., Gros, S., Diehl, M., 2017. Induction in optimal control of multiple-kite airborne

542

wind energy systems. IFAC PapersOnLine 50-1, 153-158.543

26

Loyd, M. L., 1980. Crosswind kite power. Journal of Energy (AIAA) 4, 106–111. Article No.

544

80-4075.545

Okulov, V., van Kuik, G.A., 2009. The Betz-Joukowsky limit for the maximum power

546

coeﬃcient of wind turbines. International Scientiﬁc Journal for Alternative Energy and

547

Ecology 9, 106–111.548

Payne, P. R., McCutchen, C., 1976. Self-erecting Windmill. US Patent 3,987,987.549

Rancourt, D., Bolduc-Teasdale, F., Bouchard, E. D., Anderson, M. J., Mavris, D. N., 2016.

550

Design space exploration of gyrocopter-type airborne wind turbines. Wind Energy, 195,

551

895–909.552

Schemehl, R. (Editor), 2018. Airborne Wind Energy. Singapore: Springer.553

Sørensen, J. N., 2016. General Momentum Theory for Horizontal Axis Wind Turbines.

554

Research Topics in Wind Energy, Peinke, J. (editor), Volume 4. Switzerland: Springer.555

Vander Lind, D., 2013. Analysis and ﬂight test validation of high performance airborne wind

556

turbines. In: Ahrens U., Diehl M., Schmehl R., editors. Airborne Wind Energy. Berlin

557

Heidelberg: Springer, 473–490.558

Zanon, M., Gros, S., Andersson, J., Diehl, M., 2013. Airborne wind energy based on dual

559

airfoils. IEEE Transactions on Control Systems Technology, 214, 1215-1222.560

Zanon, M., Gros, S., Meyers, J., Diehl, M., 2014. Airborne wind energy: airfoil-airmass

561

interaction. In: the 19th International Federation of Automatic Control (IFAC) World

562

Congress, South Africa, August 24-29.563

27

wind ow

kite

tether

drum and

generator ight trajectory

to the grid

(a)

wind ow

ight trajectory

drum and

motor

tether

kite

from the grid

(b)

Figure 1: Schematic view of the lift mode or ground-based generation: (a) reel-out (or power generation)

phase, (b) reel-in (or power consumption) phase. The present drawing was inspired by the graphic design

made by Daidalos Capital (2016).

28

v∞

vd

vd

vd

vd

v∞−vd

vrvw−vd

inlet

streamtube outlet

1

Figure 3: Schematic showing a wind turbine rotor subjected to a uniform wind ﬁeld of (absolute) velocity

v∞

and moving downwind at constant speed

vd

=

ev∞

, where 0

≤e≤

1. The dashed lines show a streamtube

around the rotor, which extends from far upstream to far downstream of the rotor. The ﬂow velocity at the

inlet of the streamtube is

v∞−vd

and that at the outlet is

vw−vd

. Also, the ﬂow velocity at the rotor is

represented by vr.

30

x

y

va

vc

Vrel

L

D

F

φ

φγ

θ

α

1

Figure 4: Velocity vectors and aerodynamic forces acting on the airfoil section of a crosswind kite:

va

,

vc

,

and

Vrel

represent, respectively, the axial, lateral, and total relative ﬂuid-kite velocities;

L

,

D

, and

F

are,

respectively, the lift, drag, and total aerodynamic forces;

α

is the angle of attack,

θ

is the pitch angle,

γ

is

the angle between Fand the x-axis, and φ=θ+αis the angle between Vrel and the y-axis.

31

0

0.01

0.02

0.03

0

50

100

150

200

0

5

10

15

20

25

30

0

5

10

15

20

25

σ=Ak

As

χ=CL(CL

CD)2

C(k)

pL

1

Figure 5: The kite-area-normalized coeﬃcient of the useful power (or power output) for a lift mode CKPS,

C(k)

pL

=

PL/

(1

/

2)

ρAkv3

∞

, as a function of the solidity factor,

σ

, and the aerodynamic eﬃciency,

χ

, for reel-out

ratio

e

= 1

/

3. The black polyline shows the border between acceptable (inner) and unacceptable (outer)

regions of the plot. The acceptable region is obtained by satisfying a non-negative relative wake velocity

condition, i.e. aL≤1/2; refer to equation (3).

32

0

0.01

0.02

0.03

0

50

100

150

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

σ=Ak

As

χ=CL(CL

CD)2

aL

1

Figure 6: The induction factor for a lift mode CKPS as a function of the solidity factor,

σ

, and the aerodynamic

eﬃciency,

χ

, for reel-out ratio

e

= 1

/

3. The black polyline shows the border between acceptable (inner) and

unacceptable (outer) regions of the plot. The acceptable region is obtained by satisfying a non-negative

relative wake velocity condition, i.e. aL≤1/2; refer to equation (3).

33

0

0.01

0.02

0.03

0

50

100

150

200

0

10

20

30

40

50

60

0

10

20

30

40

50

σ=Ak

As

χ=CL(CL

CD)2

C(k)

pL,loss

1

Figure 7: The kite-area-normalized coeﬃcient of the wasted power for a lift mode CKPS,

C(k)

pL,loss

=

P(k)

L,loss/

(1

/

2)

ρAkv3

∞

, as a function of the solidity factor,

σ

, and the aerodynamic eﬃciency,

χ

, for reel-out ratio

e

= 1

/

3. The black polyline shows the border between acceptable (inner) and unacceptable (outer) regions of

the plot. The acceptable region is obtained by satisfying a non-negative relative wake velocity condition, i.e.

aL≤1/2; refer to equation (3).

34

0

0.01

0.02

0.03

0

50

100

150

200

0.4

0.6

0.8

1

1.2

1.4

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

σ=Ak

As

χ=CL(CL

CD)2

κ∗

1

Figure 8: The optimal value of the ratio of on-board turbines thrust to kite drag,

κ∗

, for a drag mode CKPS

as a function of the solidity factor, σ, and the aerodynamic eﬃciency, χ.

35

0

0.01

0.02

0.03

0

50

100

150

200

0

5

10

15

20

25

30

0

5

10

15

20

25

σ=Ak

As

χ=CL(CL

CD)2

C(k)

pD

1

Figure 9: The kite-area-normalized coeﬃcient of the useful power for a drag mode CKPS,

C(k)

pD

=

PD/

(1

/

2)

ρAkv3

∞

, as a function of the solidity factor,

σ

, and the aerodynamic eﬃciency,

χ

, for optimal values of

the ratio of the on-board turbines thrust to the kite drag, κ∗.

36

0

0.01

0.02

0.03

0

50

100

150

200

0

0.05

0.1

0.15

0.2

0.25

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

σ=Ak

As

χ=CL(CL

CD)2

aD

1

Figure 10: The induction factor for a drag mode CKPS as a function of the solidity factor,

σ

, and the

aerodynamic eﬃciency,

χ

, for optimal values of the ratio of the on-board turbines thrust to the kite drag,

κ∗

. All the points in the plot are within the acceptable range

aD≤

0

.

5 which is obtained by satisfying a

non-negative wake velocity condition; refer to equation (3).

37

0

0.01

0.02

0.03

0

50

100

150

200

0

20

40

60

0

10

20

30

40

50

σ=Ak

As

χ=CL(CL

CD)2

C(k)

pD,loss

1

Figure 11: The kite-area-normalized coeﬃcient of the wasted power for a drag mode CKPS,

C(k)

pD,loss

=

P(k)

D,loss/

(1

/

2)

ρAkv3

∞

, as a function of the solidity factor,

σ

, and the aerodynamic eﬃciency,

χ

, for optimal values of

the ratio of the on-board turbines thrust to the kite drag, κ∗.

38

0

0.01

0.02

0.03

0

50

100

150

200

1

1.5

2

2.5

1

1.2

1.4

1.6

1.8

2

σ=Ak

As

χ=CL(CL

CD)2

C(k)

pD

C(k)

pL

1

Figure 12: The ratio of the kite-area-normalized useful power coeﬃcient for a drag mode CKPS to that of a lift

mode CKPS,

C(k)

pD/C(k)

pL

, as a function of the solidity factor,

σ

, and the aerodynamic eﬃciency,

χ

, for optimal

values of the ratio of the on-board turbines thrust to the kite drag,

κ∗

, for the drag mode system and reel-out

ratio

e

= 1

/

3 for the lift mode system. The black polyline shows the border between acceptable (inner) and

unacceptable (outer) regions of the plot. The acceptable region is obtained by satisfying a non-negative

relative wake velocity condition, i.e. aL≤1/2; refer to equation (3).

39

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

aL

(C(k)

pL/χ), C(s)

pL

(C(k)

pL/χ)

C(s)

pL

(C(k)

pL,max /χ) = 4/27 C(s)

pL,max = 4/27

aL0aL= 1/2

(2)

(1)

1

Figure 13: Variation of the kite-area-normalized useful power coeﬃcient divided by the aerodynamic eﬃciency,

(

C(k)

pL/χ

) (red line), and the swept-area-normalized useful power coeﬃcient,

C(s)

pL

(blue line), for a lift mode

CKPS as a function of the induction factor,

aL

, for

e

= 1

/

3. The maximum of the (

C(k)

pL/χ

) and

C(s)

pL

curves

are located at aL= 0 and aL= 1/2, respectively.

40

0 0.05 0.1 0.15 0.2 0.25 0.3

0.1

0.2

0.3

0.4

0.5

0.6

aD

(C(k)

pD/χ), C(s)

pD

(C(k)

pD/χ)

C(s)

pD

(C(k)

pD,max /χ) = 4/27

C(s)

pD,max = 16/27

aD0

aD= 1/3

(1)

(2)

1

Figure 14: Variation of the kite-area-normalized useful power coeﬃcient divided by the aerodynamic eﬃciency,

(

C(k)

pD/χ

) (red line), and the swept-area-normalized useful power coeﬃcient,

C(s)

pD

(blue line), for a drag mode

CKPS as a function of the induction factor,

aD

, for optimal values of

κ

. The maximum of the (

C(k)

pD/χ

) and

C(s)

pDcurves are located at aD= 0 and aD= 1/3, respectively.

41